Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $r \neq 0$. $n = \dfrac{r + 5}{r + 7} \times \dfrac{r^2 + 16r + 63}{-8r - 72} $
Solution: First factor the quadratic. $n = \dfrac{r + 5}{r + 7} \times \dfrac{(r + 7)(r + 9)}{-8r - 72} $ Then factor out any other terms. $n = \dfrac{r + 5}{r + 7} \times \dfrac{(r + 7)(r + 9)}{-8(r + 9)} $ Then multiply the two numerators and multiply the two denominators. $n = \dfrac{ (r + 5) \times (r + 7)(r + 9) } { (r + 7) \times -8(r + 9) } $ $n = \dfrac{ (r + 5)(r + 7)(r + 9)}{ -8(r + 7)(r + 9)} $ Notice that $(r + 9)$ and $(r + 7)$ appear in both the numerator and denominator so we can cancel them. $n = \dfrac{ (r + 5)\cancel{(r + 7)}(r + 9)}{ -8\cancel{(r + 7)}(r + 9)} $ We are dividing by $r + 7$ , so $r + 7 \neq 0$ Therefore, $r \neq -7$ $n = \dfrac{ (r + 5)\cancel{(r + 7)}\cancel{(r + 9)}}{ -8\cancel{(r + 7)}\cancel{(r + 9)}} $ We are dividing by $r + 9$ , so $r + 9 \neq 0$ Therefore, $r \neq -9$ $n = \dfrac{r + 5}{-8} $ $n = \dfrac{-(r + 5)}{8} ; \space r \neq -7 ; \space r \neq -9 $